Yes, you should switch. In the Monty Hall problem, a prize sits behind one of three doors, you pick one, and the host, who knows where the prize is, opens a different door to show a goat and offers you a swap. Staying wins 1/3 of the time. Switching wins 2/3 of the time, twice as often. The reason fits in one sentence: switching wins exactly when your first choice was wrong, and your first choice is wrong two times out of three. The open door does not change those odds. It only tells you which closed door to move to.
Almost everyone’s first answer is fifty-fifty: two doors left, one prize. Thousands of clever people once said so, in writing.
The rules that make it work
The puzzle only has an answer once the host’s behavior is pinned down, so the video states the rules before anything else:
- The prize is equally likely to be behind any of the three doors.
- The host knows where it is.
- The host always opens a door you did not choose, and it always hides a goat.
- The host always offers you the switch.
Rule 3 is the one that matters. The host is not opening a door at random; he picks one he knows is safe, and that choice carries information.
Counting the three cases
Say you choose door 1. The prize can be in three places, all equally likely.
- Prize behind door 1. The host can reveal either goat, behind door 2 or door 3. If you switch, you lose.
- Prize behind door 2. The host cannot open your door, and he cannot open door 2, so he must open door 3. If you switch, you move to door 2 and win.
- Prize behind door 3. Now he must open door 2. If you switch, you move to door 3 and win.
That is the whole proof. The video lays the three rows side by side so you can watch the tally build.
Why switching wins twice as often
Look at the cases again and ask what decides each one. Switching loses only when your first choice was right. It wins every time your first choice was wrong. When you chose, you had one chance in three of being right, and nothing the host does afterwards can reach back and change that. So switching wins two times in three.
Marilyn vos Savant, who answered the puzzle in print, suggested making it bigger to feel it:
Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you? — Marilyn vos Savant, “Ask Marilyn”, Parade
The rules really do matter. If the host does not know where the prize is and opens a door that just happens to show a goat, switching wins only half the time. The 2/3 comes from his knowledge.
Where the Monty Hall problem came from
The puzzle is named after Monty Hall, the original host of the American game show Let’s Make a Deal. The statistician Steve Selvin posed it in a letter to The American Statistician in 1975 and called it the “Monty Hall problem” in a later letter. The same mathematics had appeared even earlier, as the three prisoners problem in Martin Gardner’s “Mathematical Games” column in Scientific American in 1959.
It became famous in 1990, when a reader, Craig F. Whitaker, sent the question to vos Savant’s “Ask Marilyn” column in Parade magazine, with a car behind one door and goats behind the other two. She said to switch. About 10,000 readers wrote to the magazine, nearly 1,000 of them with PhDs, and most of them said she was wrong. She was right.
For another count that beats intuition, see four letters, four envelopes, where every letter lands in the wrong envelope more often than most people guess.
Switch: your first choice was probably wrong.
